Question

The coordinates of the vertices of quadrilateral GOLF are G(1, -1), O(-1, -4), L(-4, -2), and F(-2, 1). Prove or disprove that the quadrilateral is a square. Be sure to show all work and provide any necessary explanation.

Hint: The grid is provided for your use (optional), however, it cannot be used to prove or disprove if the shape is a square. The graph is not worth any points and will not be considered when graded.

Answer 1

After calculating the lengths of all sides, they were found to be equal, but the slopes of adjacent sides were not negative reciprocals. Therefore, GOLF is not a square.

Step-by-step explanation:

To determine if the quadrilateral GOLF with vertices G(1, -1), O(-1, -4), L(-4, -2), and F(-2, 1) is a square, we need to prove that all four sides are of equal length and that all angles are right angles.

First, calculate the distances between adjacent vertices to determine the lengths of the sides:

• GO: √[(1 - (-1))^2 + (-1 - (-4))^2] = √[2^2 + 3^2] = √[13]
• OL: √[(-1 - (-4))^2 + (-4 - (-2))^2] = √[3^2 + (-2)^2] = √[13]
• LF: √[(-4 - (-2))^2 + (-2 - 1)^2] = √[2^2 + (-3)^2] = √[13]
• FG: √[(-2 - 1)^2 + (1 -(-1))^2] = √[(-3)^2 + 2^2] = √[13]

All sides are of equal length, suggesting that GOLF could be a square. However, we must also ensure that all angles are right angles.

For quadrilateral GOLF to be a square, the product of the slopes of two adjacent sides must be -1 (indicative of right angles). Calculate the slopes:

• Slope of GO: (y2 - y1)/(x2 - x1) = (-4 - (-1))/(-1 - 1) = -3/-2 = 1.5
• Slope of OL: (-2 - (-4))/(-4 - (-1)) = 2/-3 = -2/3
• Slope of LF: (1 - (-2))/(-2 - (-4)) = 3/2 = 1.5
• Slope of FG: (-1 - 1)/1 - (-2) = -2/3

The slopes of GO and OL as well as LF and FG are not negative reciprocals of each other. Hence, these pairs are not perpendicular, proving that GOLF is not a square.